Question: Find the value of $c$ so that the polynomial $p(x)$ is divisible by $(x+2)$. $p(x) = 4x^3+cx^2+x+2$ $c=$
Answer: The following statements are equivalent: $(x+2)$ is a factor of $p(x)$ $p(x)$ is divisible by $(x+2)$ The remainder of $\dfrac{p(x)}{x+2}$ is $0$ We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, the remainder when $p(x)$ is divided by $(x+2)$, which can be rewritten as $(x-({-2}))$, is equal to $p({-2})$. We want this remainder to be equal to $0$. So let's set $p({-2})=0$ and solve this equation to find $c$. Let's plug ${x=-2}$ in $p( x) = 4 x^3+c x^2+ x+2$ and set that equal to $0$. $\begin{aligned} 4({-2})^3+c({-2})^2 + ({-2}) + 2&=0 \\\\ -32+4c+(-2)+2&=0 \\\\ -32 + 4c&=0 \\\\ 4c&=32 \\\\ c&=8 \end{aligned}$ To conclude, $c=8$